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# Week 1 Lecture Notes for Math 127 MATH127

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## Popular in Calculus I for Life and Social Science Majors

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This 5 page Class Notes was uploaded by Monica Weisenbach on Friday January 30, 2015. The Class Notes belongs to MATH127 at University of Massachusetts taught by Thurlow Cook in Spring2015. Since its upload, it has received 64 views. For similar materials see Calculus I for Life and Social Science Majors in Mathematics (M) at University of Massachusetts.

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Date Created: 01/30/15

Week 1 VVhatEsafuncHon A function is a relationship between the inputxdomain and outputyrange An input gets an output and the same one every time There is an input an output and a rule of correspondence between them fx There is only y per x but can be multiple X s per y Vertical line rule a vertical line drawn through any point of the graph should hit the graph only once Example two people can have the same bank balances on different accounts but one person cannot have two bank balance on the same account The domain is all the values of x that solve the function The range is all the values of y that solve the function UnearFunc ons Linear functions often have the equation y mxb m is the slope b is the yintercept OY and x and y are a set of coordinate points on the graph The X intercept is called the root also XO Slope is constant there is a constant inclination in linear functions The slope is discovered by taking two points on the graph X1Y1 and XY and running them through this equation riserun Y1Y X1X make sure each coordinate pair stays together vertically Pointslope form is another way to write linear equations YY1 mXX1 This form allows for both slope and a point on the line X1Y1 Pointslope form can be changed into ymxb fairly easily Begin with Y Y1mXX1 distribute the m getting YY1mXmX1 now isolate the y getting YY1mXmX1 rearrange to be YmXY1mX1 and since Y1mX1 is the equivalent of b you end up with YmXb Average Rate of Change ln linear functions this is equivalent to the slope When m gt O that s a positive increasing slope When m lt O that s a negative decreasing sope When m O that s a horizontal line Orun In the case of a vertical line that s a dividebyzero riseO error and there is no sope ln nonlinear functions the average rate of change is found by picking two points and finding the slope of a straight line between them The average rate of change goes down if the two points get closer together For example with fxx2 the slope between the points of f2 and f1 is 3 while the slope between the points of f15 and f1 is 25 Calculus looks in large part at instantaneous rates of change Getting new functions Functions can be combined added subtracted multiplied divided Note that subtraction and division are not commutable order matters while addition and multiplication are order does not matter Take fxx21 and gx2x1 fXgX or fgX X212X l X22X2 fXgx or fgx X21 2X1 X22X l l X22X fxgx or fgx X21 2X1 2X3X22X1 Now what about taking fgx or gfx fgx is the same as f2x1 Simply use gx in place of the variable in fx so fgx 2X l2 l 4X24X1 l 4X24X2 gfx is the same as gx21 Again simply use fx in place of the variable in gx so gfx 2X211 2X221 2X23 Another way to notate this is to use fgx Quadratic A quadratic function is a more complex polynomial function than a linear function Quadratics always have a degree of 2 A quadratic function is a power function whereas 2X would be an exponential func on A very simple quadratic formula would be yx2 Many quadratics will be a combination of a simple quadratic and a linear function such as y2x24x1 A formula for a quadratic would be yax2bxc The yintercept of this formula would be c The roots xintercepts can be found by factoring or the quadratic formula Quadratic roots are where i comes into play Looking at three graphs fxx21 fxx2 and fxx21 we can see that the first has two roots the middle has one root and the last has no roots The last one technically does have roots and can be factored they are just imaginary numbers see below The imaginary number system was developed by Descartes At first people were very disturbed that it couldn t be put on a number line Around 1800 people began to show imaginary numbers on a graph This method was started by Argand Solve x2 1O Solve x2O X21 XO X XO Xi1 Solve x21 X21 X Xil Trigonometric Functions First up sinx and cosx These are periodic functions with the period being 2H Sinx392 cosx Remember SOHCAHTOA sinxoppositehypotenuse cosxadjacenthypotenuse and tanxoppositeadjacent Circles have 360 degrees and 2H radians Remember 603090 and 454590 special triangles Sinx and oosx are important because they are the solutions to differential equations things like the graph of a pendulum swing V i 3

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